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## Complexity and Applications of Edge-Induced Vertex-Cuts (2006)

Citations: | 2 - 0 self |

### Citations

696 |
Network Flows
- Ahuja, Magnanti, et al.
- 1993
(Show Context)
Citation Context ... is due to Stoer and Wagner [15] and has a runtime of O(nm + n2 log n). Its generalization to hypergraphs [11,12] has a runtime of O(np + n2 log n). The interested reader is referred to other sources =-=[1,10]-=- for further mincut algorithms and their complexity. Table 2 gives an overview of the time complexity of the above considered minimum edge-cut and vertex-cut problems for weighted hypergraphs and grap... |

672 | A new approach to the maximum-flow problem
- Goldberg, Tarjan
- 1988
(Show Context)
Citation Context ...thms for computing minimums-t-edge-cuts in graphs have been presented in the literature; most of them are based on network flow techniques. One such well-known algorithm is due to Goldberg and Tarjan =-=[6]-=- and runs in time O(n 2 √ m). We will now shortly describe how to reduce in polynomial time all remaining edge-cuts and vertex-cuts of hypergraphs and graphs to s-t-edge-cuts of graphs, which results ... |

636 |
Parameterized Complexity Theory
- Flum, Grohe
- 2006
(Show Context)
Citation Context ....t. k does not increase the order of the polynomial. Parameterized complexity was initiated by Downey and Fellows in the late 1980s and has become an important branch of algorithm design and analysis =-=[3,13,5]-=-. It turned out that the distinction between tractability of type (i) and type (ii) is a robust indication of problem hardness. XP denotes the class of all parameterized decision problems which can be... |

444 | Invitation to Fixed-Parameter Algorithms.
- Niedermeier
- 2006
(Show Context)
Citation Context ....t. k does not increase the order of the polynomial. Parameterized complexity was initiated by Downey and Fellows in the late 1980s and has become an important branch of algorithm design and analysis =-=[3,13,5]-=-. It turned out that the distinction between tractability of type (i) and type (ii) is a robust indication of problem hardness. XP denotes the class of all parameterized decision problems which can be... |

403 |
Graph Algorithms
- Even
- 1979
(Show Context)
Citation Context ...edge-cut can always be reduced to n−1 computations of a minimum s-t-edge-cut. Thus, computing minimum edge-cuts in graphs can be done in time O(n 3 √ m). Moreover, by transforming vertices into edges =-=[4]-=-, we are able to reduce the minimum s-t-vertex-cut problem in a graph with n vertices and m edges to a minimum s-t-edge-cut problem in a graph with 2n vertices and n + 2m edges. Thus, we obtain an alg... |

183 | A simple min-cut algorithm
- Stoer, Wagner
- 1997
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Citation Context ...s finally mention that there exists also a very simple and efficient algorithm for computing minimum edge-cuts which is not based on network flow techniques. This algorithm is due to Stoer and Wagner =-=[15]-=- and has a runtime of O(nm + n2 log n). Its generalization to hypergraphs [11,12] has a runtime of O(np + n2 log n). The interested reader is referred to other sources [1,10] for further mincut algori... |

165 | Hypertree decompositions and tractable queries
- GOTTLOB, LEONE, et al.
- 1999
(Show Context)
Citation Context ... optimization are conceivable. Our original motivation for investigating edge-induced vertex-cuts, however, comes from the area of constraint satisfaction; in particular, from hypertree decomposition =-=[7]-=-. In hypertree decomposition, a hypergraph is transformed into clusters of hyperedges that are organized as a tree which has to satisfy several conditions. The connection to our work here is that each... |

26 |
Modeling hypergraphs by graphs with the same mincut properties
- Ihler, Wagner, et al.
- 1993
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Citation Context ...ertex-cuts of hypergraphs and graphs to s-t-edge-cuts of graphs, which results in polynomial-time algorithms for all these cut problems. Note that this does not contradict the results of Ihler et al. =-=[9]-=-, who showed that computing minimum edge-cuts in hypergraphs cannot be reduced to computing minimum edge-cuts in (undirected) graphs without using negative weights; however, in our reduction, we use d... |

24 |
An algorithm for a minimum cover of a graph
- Norman, Rabin
- 1959
(Show Context)
Citation Context ...e also NP-hardness of computing minimum edgeinduced vertex-cuts in hypergraphs. However, these proofs do not work for ordinary graphs, since SET COVER is polynomially solvable via matching algorithms =-=[14]-=- if the size of the sets is bounded by 2. On the other hand, the reductions used for showing Theorems 3 and 4 are not fpt-reductions and do not establish W[2]-hardness. 6 Applications of Edge-Induced ... |

13 | A Simple Hypergraph Min Cut Algorithm
- Wagner, Klimmek
- 1996
(Show Context)
Citation Context ... for computing minimum edge-cuts which is not based on network flow techniques. This algorithm is due to Stoer and Wagner [15] and has a runtime of O(nm + n2 log n). Its generalization to hypergraphs =-=[11,12]-=- has a runtime of O(np + n2 log n). The interested reader is referred to other sources [1,10] for further mincut algorithms and their complexity. Table 2 gives an overview of the time complexity of th... |

9 |
Graph Theory. Number 173
- Diestel
- 2005
(Show Context)
Citation Context ...uts and vertex-cuts to our new edge-induced vertex-cuts, which we will introduce in Section 4. The following definitions are straight-forward generalizations of “separating sets” of graphs defined in =-=[2]-=-. Let us start with edge-cuts: Definition 1 (Edge-Cut). Let H = (V, E) be a hypergraph and s, t ∈ V with s �= t. A set C ⊆ E is an s-t-edge-cut in H if s and t are not connected in H ′ = (V, E\C). A s... |

8 |
Multiterminal Flows in a Hypergraph
- Hu, Moerder
- 1985
(Show Context)
Citation Context ...es in E, i.e., p = � e∈E |e|. Now, we transform each hyperedge into a star, i.e., for each hyperedge we introduce a new vertex and connect this vertex with all vertices in the corresponding hyperedge =-=[8]-=-. Afterwards, the vertices corresponding to a hyperedge in the hypergraph or the vertices corresponding to vertices in the hypergraph (depending on whether we want to compute edge-cuts or vertex-cuts)... |

3 |
A fast hypergraph min-cut algorithm for circuit partitioning
- Mak, Wong
(Show Context)
Citation Context ... for computing minimum edge-cuts which is not based on network flow techniques. This algorithm is due to Stoer and Wagner [15] and has a runtime of O(nm + n2 log n). Its generalization to hypergraphs =-=[11,12]-=- has a runtime of O(np + n2 log n). The interested reader is referred to other sources [1,10] for further mincut algorithms and their complexity. Table 2 gives an overview of the time complexity of th... |

2 |
and Hanjo Täubig. Connectivity
- Kammer
- 2005
(Show Context)
Citation Context ...nd vertex-connectivity are the same in both cases, it is easy to find hypergraphs where they do not coincide. However, for ordinary graphs, there is a well-known relationship between these invariants =-=[10]-=-: κ(G) ≤ λ(G) ≤ δ(G) for all graphs G, where δ(G) is the minimal degree over all vertices in G. Note that this result does not hold for hypergraphs in general. For example, consider a hypergraph H con... |